Transformer output usually placed in tube amps, to convert speaker impedance towards tube high voltages.
In front of me now lies a solid state power amp, no-feedback design, voltage is 0---+50V, but the output is using transformer, in push-pull mode. The final transisors are mosfets, driven by small driving transformer (the designer sure likes transformer)

Big problem with the amp. When tested with pink noise, the curve is flat. Played with pop music, the sound is very good. But if played with piano sound (about 400-600hz), the sound becomes distorted, even in very low volume.

At first I suspect the driver. Trace the signal with 500hz test tone, all signals are good sinusoidal until gates of final mosfet. In the gates, the sinusiodal are good. but in the Drain of the final mosfets (which drive the final output transformer), the sinusoidal becomes distorted. Looking at the output with scope, (speaker attached), the sinusoidal in the speaker is somehow not a good sinusioidal.

How can I fix this? Is it possible that the no-good part is the final transformer itself? Oscilating at 500hz?

I think this kind of behavior is exactly what you should expect from such a nonsense design

MOSFETs show pretty non-linear transfer characteristics and when they are operated open-loop in such a flawed topology, they will add lots of asymmetric distortion to any asymmetric signal and this is potentially a source of DC applied to the transformer, that would require a *huge* gap to handle it

Note that audio waveforms are almost allways of asymmetric nature even if they are high-pass filtered to remove any residual DC. Sometimes one side has half the peak amplitude than the other side of the waveform [Look at piano waveforms with the oscilloscope]

I think the simplest way to cure that problem is to add some feedback to linearise the circuit and reduce the DC error seen by the transformer

I cannot jigsaw the core to put gap, or put C between mosfet and core. I attach the schematic, maybe it will help you guys helping me.
I think about giving RLC (like the one used in speaker to flatten impedance curve). Putting this in output windings do not help at all. Will putting in both push-pull windings will help?

Is this schematic from an audio amplifier or from a switched mode power supply?. It looks like a push-pull SMPS with its control IC and its pulse transformer

The only way to get this topology to work properly is to ensure no DC at all is applied to the transformers. In SMPSs applications this is acomplished by using the same duty cycle for both sides and adding some dead time to allow small errors to self compensate. In audio applications you have to sense the voltage applied to the transformer, integrate it and compensate for DC errors introduced by non-balanced distortion components [removing bias to make it work in class B would also help in reducing DC applied to the transformer, this is the same as adding dead time on SMPS applications, but distortion wiil increase a lot, specially since there is no feedback at all]